Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

On the Basic Convex Polytopes and n-Balls in Complex Dimensions

Version 1 : Received: 5 September 2022 / Approved: 6 September 2022 / Online: 6 September 2022 (10:13:02 CEST)
Version 2 : Received: 13 September 2022 / Approved: 14 September 2022 / Online: 14 September 2022 (08:37:54 CEST)
Version 3 : Received: 16 September 2022 / Approved: 19 September 2022 / Online: 19 September 2022 (10:21:05 CEST)
Version 4 : Received: 6 October 2022 / Approved: 7 October 2022 / Online: 7 October 2022 (08:30:43 CEST)
Version 5 : Received: 24 December 2022 / Approved: 26 December 2022 / Online: 26 December 2022 (11:08:05 CET)
Version 6 : Received: 24 February 2023 / Approved: 24 February 2023 / Online: 24 February 2023 (15:41:43 CET)

A peer-reviewed article of this Preprint also exists.

Łukaszyk, S.; Tomski, A. Omnidimensional Convex Polytopes. Symmetry 2023, 15, 755. https://doi.org/10.3390/sym15030755 Łukaszyk, S.; Tomski, A. Omnidimensional Convex Polytopes. Symmetry 2023, 15, 755. https://doi.org/10.3390/sym15030755

Abstract

This paper extends the findings of the prior research concerning n-balls, regular n-simplices, and n-orthoplices in real dimensions using recurrence relations that removed the indefiniteness present in known formulas. The main result of this paper is the proof that these recurrence relations are continuous for complex n, whereas in the indefinite points their values are given in the sense of a limit of a function. It is shown that the volume of an n-simplex is a bivalued function for n < 0, and thus the surfaces of n-simplices and n-orthoplices are also bivalued functions for n < 1. Applications of these formulas to these omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects in negative, real dimensions. In particular for 0 < n < 1 the volumes of the omnidimensional polytopes are larger than volumes of circumscribing n-balls, while their volumes and surfaces are smaller than volumes of inscribed n-balls.

Keywords

regular basic convex polytopes; negative dimensions; fractal dimensions; complex dimensions; circumscribed and inscribed polytopes

Subject

Computer Science and Mathematics, Geometry and Topology

Comments (1)

Comment 1
Received: 19 September 2022
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author
Comment: 2. Corrected proof that the volume of a regular n-simplex circumscribed about n-ball is real for = −(2k + 1)/2 and natural k.
3. Corrected proof that the surface of a regular n-simplex circumscribed about n-ball is real for = −(2k + 1)/2 and natural k.
3. Proof that the volume of a regular n-simplex circumscribed about n-ball is complex with the real part being equal to the imaginary part up to a modulus for = −(2k + 3)/4 and natural k.
4. Proof that the surface of a regular n-simplex circumscribed about n-ball is complex with the real part being equal to the imaginary part up to a modulus for = −(2k + 3)/4 and natural k
.
5. Corrections of various typos and inaccuracies.
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